The present invention relates to a method of constructing surface element layers for improving shape of hexahedral mesh used in finite element analysis and a method of constructing hexahedral mesh using the method of constructing surface element layers.
The finite element analysis is widely used for predicting safety of various structures, analyzing engineering problems associated with differential equation such as distribution analysis of electromagnetic field and the like and metal forming processes such as extrusion and forging.
During the finite element analysis, locally concentrated large deformation occurs around corners of an object to be analyzed. Such locally concentrated large deformation makes determinant of transformation matrix of an element negative, so that the analysis can be impossible. When a mesh element is severely distorted and the negative determinant arises, a new mesh having well-shaped elements has to be constructed to replace the degenerated old mesh. Therefore, an automatic mesh-reconstructing method, hereinafter referred to as remeshing, is required for a smooth and rapid finite element analysis.
Of automatic 3-dimentional mesh construction methods, tetrahedral mesh construction methods such as Delauney algorithm and Octree techniques have been widely used because the methods enable very robust and near-optimal mesh to be constructed.
On the other hand, hexahedral mesh construction methods are preferred because the methods improve accuracy of the analysis. The hexahedral mesh construction methods mainly based on a mapping method have disadvantages that a complex-shaped 3-dimentional region to be meshed has to be semi-automatically divided into several simple sub-regions in which mesh can be constructed and that ineffective mesh can be constructed because local density of mesh cannot be controlled.
A grid-based approach of the conventional hexahedral mesh construction methods enables a hexahedral mesh to be automatically constructed without semi-automatic process such as manual dividing into sub-regions.
Technique of constructing a mesh using a grid-based approach is described in a paper entitled xe2x80x9cApplication of FEM to Prediction of Microstructure in Hot Forming of Metalsxe2x80x9d, written by R. Kopp. K. Karhausen and R. Schneiders, and published in Proc. of 4th ICTP, pp. 1203-1211 in 1993.
However, because many surface elements are severely distorted in the technique of constructing a mesh using the grid-based approach, this technique has a disadvantage that accurate prediction cannot be performed in analysis.
A technique of constructing a mesh by use of a grid-based approach, dividing ill-shaped elements into several sorts and repositioning corresponding elements according to fixed regulation is disclosed in a paper entitled xe2x80x9cAutomatic Mesh Generation from CAD on Vector-parallel and Massively Parallel Supercomputersxe2x80x9d, written by R. Taghavi, and published in Report of Cray Co. in 1994.
However, such technique has disadvantages that it is suitable only for large-scaled mesh and that the ill-shaped elements still remain.
A method of constructing mesh elements in space between a internal regular grid and a boundary surface of region to be meshed when constructing mesh using a grid-based approach is disclosed in a paper entitled xe2x80x9c3-D Simulation of metal Forming Processes with Automatic Mesh Generation Automatic Meshxe2x80x9d, written by A. E. Tekkaya and S. Kavakli, and published in Steel Research, Vol.66. pp.377-383 in 1995.
Such method has an advantage that relatively well-shaped elements can be constructed. However, such method has disadvantages that it is difficult to reflect geometrical characteristics of the region boundary and elements construction at the region boundary must be performed by only one method.
The grid-based mesh construction methods described above, unlike the conventional methods based on a mapping method, can automatically construct a mesh consisting of hexahedral mesh elements without dividing the whole region into sub-regions having several simple shapes. However, such methods have a disadvantage that excessive distortions of the surface elements can occur.
Therefore, the present invention is made in order to solve the conventional problems described above.
An object of the present invention is to provide a method of constructing surface element layers of a hexahedral mesh used in finite element analysis, capable of improving the shape of boundary elements of the hexahedral mesh and improving accuracy of prediction in analysis by constructing surface element layers and repositioning nodes of the surface elements.
Another object of the present invention is to provide a method of constructing a hexahedral mesh using the above method of constructing surface element layers for improving shape qualities of severely-distorted boundary surface elements.
The above objects can be accomplished by a method of constructing surface element layers of a hexahedral mesh used in finite element analysis, the method comprising: a step of constructing imaginary thin surface element layers on a boundary surface of a mesh; and a step of performing a mesh smoothing on said imaginary thin surface element layers.
The step of constructing said imaginary thin surface element layers is performed by obtaining offset vectors outward at each node on faces, edges and vertexes of said boundary surface of said mesh, positioning new nodes at positions offset in directions of said obtained offset vectors and then assigning connectivity of said new nodes for new elements.
Also, the step of performing a mesh smoothing is performed by repositioning node on vertex, node on edge and node on face of boundary surface of said imaginary thin surface element layers and internal node of said imaginary thin surface element layers, respectively.
It is preferable that the repositioning of nodes is performed in order of repositioning nodes on the edge, repositioning nodes on the face and repositioning the internal nodes, in a state that positions of nodes on the vertex are fixed.
It is more preferable that the offset vector at node on the edge of the boundary surface of the mesh is obtained by using two offset vectors obtained in each average direction of two faces adjacent to each edge.
It is still more preferable that the offset vector at node on the vertex of the boundary surface of the mesh is obtained by using three offset vectors obtained in each average direction of three faces adjacent to each vertex.
Also, the above objects can be accomplished by a method of constructing a hexahedral mesh used in finite element analysis is provided, the method comprising: a step of constructing a core mesh by superimposing a regular grid on a region to be meshed and removing external elements and external nodes of said region to be meshed; a step of amending said core mesh to have a boundary shape similar to that of said region to be meshed by repositioning nodes on the boundary of said core mesh; a step of constructing imaginary thin surface element layers on a boundary surface of said amended core mesh; and a step of performing a mesh smoothing on said imaginary thin surface element layers.
The step of constructing said imaginary thin surface element layers is performed by obtaining offset vectors outward at each node on face, edge and vertex of said boundary surface of said amended core mesh, positioning new nodes at positions offset in directions of said obtained offset vectors and then assigning connectivity of said new nodes for new elements.
Also, the step of performing a mesh smoothing is performed by repositioning node on vertex, nodes on edge and nodes on face of boundary surface of said imaginary thin surface element layers and internal nodes of said imaginary thin surface element layers, respectively.